Subgroups — C₄₁×#60144 #23836345

Order 10496.

2778 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#568	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₁×#532	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2935	C₂	1	8	yes	maximal, complemented
4	D₄×M₁₆×C₄₁	C₂	1	8	yes	maximal, complemented
5	C₄₁×#2932	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2931	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2879	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₄₁	C₂²	1	2	yes	—
11	C₄₁×#568	—	2	16	no	—
12	C₂×#131×C₄₁	C₂²	1	8	yes	complemented
13	C₂×#131×C₄₁	C₂²	1	8	yes	complemented
14	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₄₁	C₂²	1	8	yes	—
16	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#532	C₄	1	2	yes	—
19	C₄₁×#532	C₄	1	4	yes	—
20	C₄₁×#532	C₄	1	2	yes	—
21	C₄₁×#532	C₂²	1	2	yes	—
22	C₄₁×#532	C₂²	1	2	yes	—
23	C₄₁×#532	C₂²	1	4	yes	—
24	C₂×C₄⋊Q₈×C₄₁	C₄	1	1	yes	characteristic
25	C₂×#127×C₄₁	C₄	1	1	yes	characteristic
26	C₂×#124×C₄₁	C₄	1	2	yes	—
27	C₂×#121×C₄₁	C₄	1	4	yes	—
28	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
29	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#436	C₂²	1	16	yes	complemented
31	C₄₁×#435	C₂²	1	32	yes	complemented
32	C₈×D₄×C₄₁	C₂²	1	16	yes	complemented
33	C₄₁×#423	C₂²	1	4	yes	—
34	C₄₁×#423	C₂²	1	2	yes	complemented
35	C₄₁×#423	C₂²	1	2	yes	complemented
36	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	complemented
37	C₄₁×#408	C₂²	1	8	yes	complemented
38	C₄₁×#408	C₂²	1	8	yes	complemented
39	C₄₁×#407	C₂²	1	8	yes	—
40	C₄×M₁₆×C₄₁	C₂²	1	4	yes	—
41	C₄×M₁₆×C₄₁	C₂²	1	4	yes	complemented
42	C₂×#97×C₄₁	C₂²	1	2	yes	complemented
43	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	complemented
44	C₂×#41×C₄₁	—	2	8	no	—
45	C₂×#41×C₄₁	—	2	8	no	—
46	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
47	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
48	C₂×#41×C₄₁	C₂³	1	4	yes	—
49	C₂×#41×C₄₁	C₂³	1	4	yes	—
50	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
51	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
52	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
53	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
54	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
55	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
56	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
57	#131×C₄₁	—	2	32	no	—
58	#131×C₄₁	—	2	32	no	—
59	C₂×M₁₆×C₄₁	—	2	8	no	—
60	C₂×M₁₆×C₄₁	—	2	4	no	—
61	C₂×M₁₆×C₄₁	—	2	32	no	—
62	C₂×M₁₆×C₄₁	D₄	1	8	yes	complemented
63	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
64	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
65	C₂×M₁₆×C₄₁	C₂³	1	16	yes	—
66	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
71	C₂²×C₈×C₄₁	C₂³	1	16	yes	—
72	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
73	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
75	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
76	C₄⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
77	#127×C₄₁	C₂×C₄	1	2	yes	—
78	#127×C₄₁	C₂×C₄	1	4	yes	—
79	#127×C₄₁	C₂×C₄	1	2	yes	—
80	#124×C₄₁	C₂×C₄	1	4	yes	—
81	#124×C₄₁	C₂×C₄	1	4	yes	—
82	#124×C₄₁	C₂×C₄	1	8	yes	—
83	#121×C₄₁	C₂×C₄	1	8	yes	—
84	#121×C₄₁	C₂×C₄	1	8	yes	—
85	#121×C₄₁	C₂×C₄	1	8	yes	—
86	#121×C₄₁	C₂×C₄	1	8	yes	—
87	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
88	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
89	C₄×D₄×C₄₁	C₂³	1	8	yes	—
90	C₄×D₄×C₄₁	C₂³	1	8	yes	—
91	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
92	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
93	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
97	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
99	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
102	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
103	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
104	C₂²⋊C₈×C₄₁	C₂³	1	32	yes	—
105	#97×C₄₁	C₂³	1	8	yes	—
106	C₄×C₈×C₄₁	C₂³	1	8	yes	—
107	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
108	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
109	#41×C₄₁	—	2	16	no	—
110	#41×C₄₁	—	2	16	no	—
111	#41×C₄₁	—	2	8	no	—
112	#41×C₄₁	—	2	8	no	—
113	#41×C₄₁	—	2	8	no	—
114	#41×C₄₁	—	2	8	no	—
115	C₂×Q₈×C₄₁	—	2	8	no	—
116	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
117	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
118	C₂×D₄×C₄₁	—	2	16	no	—
119	C₂×D₄×C₄₁	—	2	8	no	—
120	C₂×D₄×C₄₁	—	4	8	no	—
121	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
122	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
123	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
125	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
126	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
127	C₂²×C₄×C₄₁	—	2	8	no	—
128	C₂²×C₄×C₄₁	—	2	8	no	—
129	C₂²×C₄×C₄₁	—	4	4	no	—
130	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
135	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₄₁	#41	1	4	yes	—
143	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
144	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
147	M₁₆×C₄₁	—	2	8	no	—
148	M₁₆×C₄₁	—	2	8	no	—
149	M₁₆×C₄₁	—	2	4	no	—
150	M₁₆×C₄₁	—	2	4	no	—
151	M₁₆×C₄₁	—	2	32	no	—
152	M₁₆×C₄₁	C₂×D₄	1	8	yes	—
153	M₁₆×C₄₁	C₂×D₄	1	8	yes	complemented
154	C₂×C₈×C₄₁	—	2	8	no	—
155	C₂×C₈×C₄₁	—	2	4	no	—
156	C₂×C₈×C₄₁	—	2	32	no	—
157	C₂×C₈×C₄₁	C₂×D₄	1	8	yes	complemented
158	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
159	C₂×C₈×C₄₁	C₂⁴	1	8	yes	—
160	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
161	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
162	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
163	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
164	C₂²⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
165	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
166	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
167	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
170	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
172	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₄₁	C₂⁴	1	2	yes	—
174	C₄²×C₄₁	C₂⁴	1	2	yes	—
175	C₂³×C₄₁	—	2	8	no	—
176	C₂³×C₄₁	—	4	2	no	—
177	C₂³×C₄₁	—	4	8	no	—
178	C₂³×C₄₁	C₄×D₄	1	4	yes	—
179	C₂³×C₄₁	C₂³×C₄	1	8	yes	—
180	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
181	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
182	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
183	Q₈×C₄₁	—	2	8	no	—
184	Q₈×C₄₁	—	2	8	no	—
185	D₄×C₄₁	—	2	16	no	—
186	D₄×C₄₁	—	2	16	no	—
187	D₄×C₄₁	—	2	8	no	—
188	D₄×C₄₁	—	2	8	no	—
189	D₄×C₄₁	—	2	16	no	—
190	D₄×C₄₁	—	4	8	no	—
191	C₂×C₄×C₄₁	—	2	8	no	—
192	C₂×C₄×C₄₁	—	2	16	no	—
193	C₂×C₄×C₄₁	—	2	4	no	—
194	C₂×C₄×C₄₁	—	2	4	no	—
195	C₂×C₄×C₄₁	—	2	4	no	—
196	C₂×C₄×C₄₁	—	2	8	no	—
197	C₂×C₄×C₄₁	—	2	8	no	—
198	C₂×C₄×C₄₁	—	4	2	no	—
199	C₂×C₄×C₄₁	—	4	2	no	—
200	C₂×C₄×C₄₁	—	4	2	no	—
201	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
202	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
203	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
204	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
207	C₂×C₄×C₄₁	C₂³×C₄	1	8	yes	—
208	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
212	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
216	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
217	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
218	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
219	C₈×C₄₁	—	2	8	no	—
220	C₈×C₄₁	—	2	4	no	—
221	C₈×C₄₁	C₂²×D₄	1	8	yes	—
222	#60144	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₄₁	—	2	2	no	—
224	C₂²×C₄₁	—	2	2	no	—
225	C₂²×C₄₁	—	2	2	no	—
226	C₂²×C₄₁	—	2	8	no	—
227	C₂²×C₄₁	—	2	16	no	—
228	C₂²×C₄₁	—	4	4	no	—
229	C₂²×C₄₁	—	8	8	no	—
230	C₂²×C₄₁	C₂×C₄×D₄	1	2	yes	—
231	C₂²×C₄₁	C₂×C₄×D₄	1	2	yes	—
232	C₂²×C₄₁	C₂×C₄×D₄	1	2	yes	—
233	C₂²×C₄₁	C₂⁴×C₄	1	1	yes	characteristic
234	C₄×C₄₁	—	2	8	no	—
235	C₄×C₄₁	—	2	2	no	—
236	C₄×C₄₁	—	2	2	no	—
237	C₄×C₄₁	—	2	2	no	—
238	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
239	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
240	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
241	C₄×C₄₁	#568	1	4	yes	complemented
242	C₄×C₄₁	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₄₁	C₂²×#41	1	1	yes	characteristic
244	C₂×#568	C₂×C₄₁	1	2	yes	complemented
245	C₂×#532	C₂×C₄₁	1	1	yes	characteristic
246	#2935	C₂×C₄₁	1	8	yes	complemented
247	D₄×M₁₆	C₂×C₄₁	1	8	yes	complemented
248	#2932	C₂×C₄₁	1	4	yes	complemented
249	#2931	C₂×C₄₁	1	4	yes	complemented
250	C₂×#423	C₂×C₄₁	1	1	yes	characteristic, complemented
251	#2879	C₂×C₄₁	1	2	yes	complemented
252	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
253	C₂×C₄₁	—	2	2	no	—
254	C₂×C₄₁	—	2	2	no	—
255	C₂×C₄₁	—	2	2	no	—
256	C₂×C₄₁	—	4	8	no	—
257	C₂×C₄₁	C₂²×C₄×D₄	1	1	yes	characteristic
258	C₂×C₄₁	C₂×#568	1	2	yes	—
259	C₂²×#41	C₂²×C₄₁	1	2	yes	—
260	#568	—	2	16	no	—
261	C₂×#131	C₂²×C₄₁	1	8	yes	complemented
262	C₂×#131	C₂²×C₄₁	1	8	yes	complemented
263	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
264	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
265	C₂²×M₁₆	C₂²×C₄₁	1	8	yes	—
266	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
267	#532	C₄×C₄₁	1	2	yes	—
268	#532	C₄×C₄₁	1	4	yes	—
269	#532	C₄×C₄₁	1	2	yes	—
270	#532	C₂²×C₄₁	1	2	yes	—
271	#532	C₂²×C₄₁	1	2	yes	—
272	#532	C₂²×C₄₁	1	4	yes	—
273	C₂×C₄⋊Q₈	C₄×C₄₁	1	1	yes	characteristic
274	C₂×#127	C₄×C₄₁	1	1	yes	characteristic
275	C₂×#124	C₄×C₄₁	1	2	yes	—
276	C₂×#121	C₄×C₄₁	1	4	yes	—
277	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
278	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
279	#436	C₂²×C₄₁	1	16	yes	complemented
280	#435	C₂²×C₄₁	1	32	yes	complemented
281	C₈×D₄	C₂²×C₄₁	1	16	yes	complemented
282	#423	C₂²×C₄₁	1	4	yes	—
283	#423	C₂²×C₄₁	1	2	yes	complemented
284	#423	C₂²×C₄₁	1	2	yes	complemented
285	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	complemented
286	#408	C₂²×C₄₁	1	8	yes	complemented
287	#408	C₂²×C₄₁	1	8	yes	complemented
288	#407	C₂²×C₄₁	1	8	yes	—
289	C₄×M₁₆	C₂²×C₄₁	1	4	yes	—
290	C₄×M₁₆	C₂²×C₄₁	1	4	yes	complemented
291	C₂×#97	C₂²×C₄₁	1	2	yes	complemented
292	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	complemented
293	C₄₁	#60144	1	1	yes	characteristic, direct product factor, Sylow
294	C₂×#41	—	2	8	no	—
295	C₂×#41	—	2	8	no	—
296	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
297	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
298	C₂×#41	C₂³×C₄₁	1	4	yes	—
299	C₂×#41	C₂³×C₄₁	1	4	yes	—
300	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
301	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
302	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
303	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
304	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
305	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
306	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
307	#131	—	2	32	no	—
308	#131	—	2	32	no	—
309	C₂×M₁₆	—	2	8	no	—
310	C₂×M₁₆	—	2	4	no	—
311	C₂×M₁₆	—	2	32	no	—
312	C₂×M₁₆	D₄×C₄₁	1	8	yes	complemented
313	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
314	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
315	C₂×M₁₆	C₂³×C₄₁	1	16	yes	—
316	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
317	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
318	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
319	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
320	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
321	C₂²×C₈	C₂³×C₄₁	1	16	yes	—
322	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
323	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
324	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
325	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
326	C₄⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
327	#127	C₂×C₄×C₄₁	1	2	yes	—
328	#127	C₂×C₄×C₄₁	1	4	yes	—
329	#127	C₂×C₄×C₄₁	1	2	yes	—
330	#124	C₂×C₄×C₄₁	1	4	yes	—
331	#124	C₂×C₄×C₄₁	1	4	yes	—
332	#124	C₂×C₄×C₄₁	1	8	yes	—
333	#121	C₂×C₄×C₄₁	1	8	yes	—
334	#121	C₂×C₄×C₄₁	1	8	yes	—
335	#121	C₂×C₄×C₄₁	1	8	yes	—
336	#121	C₂×C₄×C₄₁	1	8	yes	—
337	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
338	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
339	C₄×D₄	C₂³×C₄₁	1	8	yes	—
340	C₄×D₄	C₂³×C₄₁	1	8	yes	—
341	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
342	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
343	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
344	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
345	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
346	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
347	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
349	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
352	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
353	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
354	C₂²⋊C₈	C₂³×C₄₁	1	32	yes	—
355	#97	C₂³×C₄₁	1	8	yes	—
356	C₄×C₈	C₂³×C₄₁	1	8	yes	—
357	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
358	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
359	#41	—	2	16	no	—
360	#41	—	2	16	no	—
361	#41	—	2	8	no	—
362	#41	—	2	8	no	—
363	#41	—	2	8	no	—
364	#41	—	2	8	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
367	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
368	C₂×D₄	—	2	16	no	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	4	8	no	—
371	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
372	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
373	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
374	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
375	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
376	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	4	4	no	—
380	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
381	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
382	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
383	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
384	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
385	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
386	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
387	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
388	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
389	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
390	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
392	C₂²×C₄	#41×C₄₁	1	4	yes	—
393	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
394	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
397	M₁₆	—	2	8	no	—
398	M₁₆	—	2	8	no	—
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	4	no	—
401	M₁₆	—	2	32	no	—
402	M₁₆	C₂×D₄×C₄₁	1	8	yes	—
403	M₁₆	C₂×D₄×C₄₁	1	8	yes	complemented
404	C₂×C₈	—	2	8	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	32	no	—
407	C₂×C₈	C₂×D₄×C₄₁	1	8	yes	complemented
408	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
409	C₂×C₈	C₂⁴×C₄₁	1	8	yes	—
410	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
411	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
412	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
413	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
414	C₂²⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
415	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
416	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
417	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
418	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
419	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
420	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
421	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
422	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
423	C₄²	C₂⁴×C₄₁	1	2	yes	—
424	C₄²	C₂⁴×C₄₁	1	2	yes	—
425	C₂³	—	2	8	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	8	no	—
428	C₂³	C₄×D₄×C₄₁	1	4	yes	—
429	C₂³	C₂³×C₄×C₄₁	1	8	yes	—
430	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
431	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
432	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
433	Q₈	—	2	8	no	—
434	Q₈	—	2	8	no	—
435	D₄	—	2	16	no	—
436	D₄	—	2	16	no	—
437	D₄	—	2	8	no	—
438	D₄	—	2	8	no	—
439	D₄	—	2	16	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	16	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
452	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
453	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
454	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
455	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
457	C₂×C₄	C₂³×C₄×C₄₁	1	8	yes	—
458	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
459	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
460	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
461	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
465	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
466	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
467	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
468	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
469	C₈	—	2	8	no	—
470	C₈	—	2	4	no	—
471	C₈	C₂²×D₄×C₄₁	1	8	yes	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	16	no	—
476	C₂²	—	2	8	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	8	8	no	—
479	C₂²	C₂×C₄×D₄×C₄₁	1	2	yes	—
480	C₂²	C₂×C₄×D₄×C₄₁	1	2	yes	—
481	C₂²	C₂×C₄×D₄×C₄₁	1	2	yes	—
482	C₂²	C₂⁴×C₄×C₄₁	1	1	yes	characteristic
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	8	no	—
487	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
488	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
489	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
490	C₄	C₄₁×#568	1	4	yes	complemented
491	C₄	C₂³×D₄×C₄₁	1	1	yes	characteristic
492	C₄	C₂²×#41×C₄₁	1	1	yes	characteristic
493	C₂	—	2	2	no	—
494	C₂	—	2	2	no	—
495	C₂	—	2	2	no	—
496	C₂	—	4	8	no	—
497	C₂	C₂²×C₄×D₄×C₄₁	1	1	yes	characteristic
498	C₂	C₂×C₄₁×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₄₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
socle	C₂²×C₄₁	C₂⁴×C₄
prime radical	C₂×C₄₁×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₄₁×#568	C₂	semidirect
3. C₄₁×#2935	C₂	semidirect
4. D₄×M₁₆×C₄₁	C₂	semidirect
5. C₄₁×#2932	C₂	semidirect
6. C₄₁×#2931	C₂	semidirect
7. C₂×C₄₁×#423	C₂	semidirect
8. C₄₁×#2879	C₂	semidirect
9. C₂×C₄×M₁₆×C₄₁	C₂	semidirect
12. C₂×#131×C₄₁	C₂²	semidirect
13. C₂×#131×C₄₁	C₂²	semidirect
30. C₄₁×#436	C₂²	semidirect
31. C₄₁×#435	C₂²	semidirect
32. C₈×D₄×C₄₁	C₂²	semidirect
34. C₄₁×#423	C₂²	semidirect
35. C₄₁×#423	C₂²	semidirect
36. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
37. C₄₁×#408	C₂²	semidirect
38. C₄₁×#408	C₂²	semidirect
41. C₄×M₁₆×C₄₁	C₂²	semidirect
42. C₂×#97×C₄₁	C₂²	semidirect
43. C₂×C₄×C₈×C₄₁	C₂²	semidirect
62. C₂×M₁₆×C₄₁	D₄	semidirect
153. M₁₆×C₄₁	C₂×D₄	semidirect
157. C₂×C₈×C₄₁	C₂×D₄	semidirect
222. #60144	C₄₁	direct (N × H)
241. C₄×C₄₁	#568	semidirect
244. C₂×#568	C₂×C₄₁	semidirect
246. #2935	C₂×C₄₁	semidirect
247. D₄×M₁₆	C₂×C₄₁	semidirect
248. #2932	C₂×C₄₁	semidirect
249. #2931	C₂×C₄₁	semidirect
250. C₂×#423	C₂×C₄₁	semidirect
251. #2879	C₂×C₄₁	semidirect
252. C₂×C₄×M₁₆	C₂×C₄₁	semidirect
261. C₂×#131	C₂²×C₄₁	semidirect
262. C₂×#131	C₂²×C₄₁	semidirect
279. #436	C₂²×C₄₁	semidirect
280. #435	C₂²×C₄₁	semidirect
281. C₈×D₄	C₂²×C₄₁	semidirect
283. #423	C₂²×C₄₁	semidirect
284. #423	C₂²×C₄₁	semidirect
285. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
286. #408	C₂²×C₄₁	semidirect
287. #408	C₂²×C₄₁	semidirect
290. C₄×M₁₆	C₂²×C₄₁	semidirect
291. C₂×#97	C₂²×C₄₁	semidirect
292. C₂×C₄×C₈	C₂²×C₄₁	semidirect
293. C₄₁	#60144	direct (N × H)
312. C₂×M₁₆	D₄×C₄₁	semidirect
403. M₁₆	C₂×D₄×C₄₁	semidirect
407. C₂×C₈	C₂×D₄×C₄₁	semidirect
490. C₄	C₄₁×#568	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄₁×#568	2
C₂×C₄₁×#532	2
C₄₁×#2935	2
D₄×M₁₆×C₄₁	2
C₄₁×#2932	2
C₄₁×#2931	2
C₂×C₄₁×#423	2
C₄₁×#2879	2
C₂×C₄×M₁₆×C₄₁	2
#60144	41

Subgroups — C41×#60144 #23836345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#60144 #23836345